All the complex roots of $(z + 1)^5 = 32z^5,$ when plotted in the complex plane, lie on a circle.  Find the radius of this circle.
Explanation: Taking the absolute value of both sides, we get $|(z + 1)^5| = |32z^5|.$  Then
\[|z + 1|^5 = 32|z|^5,\]so $|z + 1| = 2|z|.$  Hence, $|z + 1|^2 = 4|z|^2.$

Let $z = x + yi,$ where $x$ and $y$ are real numbers.  Then
\[|x + yi + 1|^2 = 4|x + yi|^2,\]which becomes
\[(x + 1)^2 + y^2 = 4(x^2 + y^2).\]This simplifies to
\[3x^2 - 2x + 3y^2 + 1 = 0.\]Completing the square, we get
\[\left( x - \frac{1}{3} \right)^2 + y^2 = \left( \frac{2}{3} \right)^2.\]Thus, the radius of the circle is $\boxed{\frac{2}{3}}.$